Study on Transformation Group Theory and Equivariant K-theory

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[1] The classical Borsuk-Ulam theorem is only concerned with equivariant maps between spheres with the antipodal action of the cyclic group of order 2. In our study we generalize this to equivariant maps between unit spheres of unitary representations of a compact abelian group. We obtain a necessary and sufficient condition for the existence of such equivariant maps in terms of the Euler classes of unit spheres, and also obtain the result concerning with the degrees of equivariant maps. The method depends heavily on the equivariant K-theory.[II] Cuttings and pastings of manifolds lead to the so-called SK-groups of manifolds. Various kinds of such groups are found in the literature. In our study we investigate the relation among them, and show that the following three SK-groups are essentially isomorphic to each other : (i) the SK-group of pairs of manifolds, (ii) the SK-group of manifolds with boundary, and (iii) the SK-group of manifolds with involution.